Abstract

The method of excess fractions (EF) is well established to resolve the fringe order ambiguity generated in interferometric detection. Despite this background, multiwavelength interferometric absolute long distance measurements have only been reported with varying degrees of success. In this paper we present a theoretical model that can predict the unambiguous measurement range in EF based on the selected measurement wavelengths and phase noise. It is shown that beat wavelength solutions are a subset of this theoretical model. The performance of EF, for a given phase noise, is shown to be equivalent to beat techniques but offers many alternative sets of measurement wavelengths and therefore EF offer significantly greater flexibility in experimental design.

© 2011 Optical Society of America

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2010

K. Falaggis, D. P. Towers, and C. E. Towers, “Theoretical analysis and optimisation of the method of excess fractions for long-range metrology,” Proc. SPIE 7790, 779009 (2010).
[CrossRef]

2009

2008

2006

2004

J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29, 1153–1155 (2004).
[CrossRef] [PubMed]

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Generalized multi-frequency selection for full-field interferometric shape measurement,” Proc. SPIE 5502, 406–409 (2004).
[CrossRef]

2003

2002

S. Lévéque, Y. Salvadé, R. Dandliker, and O. Scherler,“High-accuracy laser metrology enhances the VLTI,” Laser Focus World 38, 101–104 (2002).

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324(2002).
[CrossRef]

2001

J. Zhong and Y. Zhang, “Absolute phase-measurement technique based on number theory in multifrequency grating projection profilometry,” Appl. Opt. 40, 492–500(2001).
[CrossRef]

M. Loefdahl and H. Eriksson, “An algorithm for resolving 2pi ambiguities in interferometric measurements by use of multiple wavelength,” Opt. Eng. 40, 984–990 (2001).
[CrossRef]

1999

1997

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

J.-L. Li, H.-J. Su, and X.-Y. Su, “Two-frequency grating used in phase-measuring profilometry,” Appl. Opt. 36, 277–280(1997).
[CrossRef] [PubMed]

Y. Surrel, “Two-step temporal phase unwrapping in profilometry” Proc. SPIE 3098, 271–282 (1997).
[CrossRef]

J. M. Huntley and H. O. Saldner, “Error-reduction methods for shape measurement by temporal phase unwrapping,” J. Opt. Soc. Am. A 14, 3188–3196 (1997).
[CrossRef]

1996

W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

1994

1991

V. I. Gushov and Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
[CrossRef]

1988

K. Creath, “Phase measurement interferometry techniques,” Progress in Optics 26, 350–393 (1988).
[CrossRef]

1984

1982

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry ,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

1977

1971

1966

K. E. Gillilland, H. D. Cook, K. D. Mielenz, and R. B. Stephens, “Use of a laser for length measurement by fringe counting,” Metrologia 2, 95–98 (1966).
[CrossRef]

1898

R. Benoît, “Application des phénomènes d'interférence à des déterminations métrologiques,” Phys. Radium 7, 57–68 (1898).

1895

A. A. Michelson and J. R. Benoit, “Détermination expérimentale de la valeur du mètre en longueurs d'ondes lumineuses,” Trav. Et Mem. Bur. Int. Poids es Mes. 11, 1–42 (1895).

1889

A. Hurwitz, “Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen,” Acta Math. 12, 367–405(1889).
[CrossRef]

Absil, O.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Ahn, S.-W.

Akeson, R. L.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Andra, P.

W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Bally, J.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Beichman, C. A.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Benoit, J. R.

A. A. Michelson and J. R. Benoit, “Détermination expérimentale de la valeur du mètre en longueurs d'ondes lumineuses,” Trav. Et Mem. Bur. Int. Poids es Mes. 11, 1–42 (1895).

Benoît, R.

R. Benoît, “Application des phénomènes d'interférence à des déterminations métrologiques,” Phys. Radium 7, 57–68 (1898).

Bordé, P.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Born, M.

M. Born and E. Wolf, Principles of Optics, (Cambridge University, 2006), Chap. 7.6.

Bothe, T.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324(2002).
[CrossRef]

Burger, J. P.

Burke, J.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324(2002).
[CrossRef]

Bustraan, K.

Cassaing, F.

Chang, Y.

Chen, W.

Cheng, Y.-Yen

Cook, H. D.

K. E. Gillilland, H. D. Cook, K. D. Mielenz, and R. B. Stephens, “Use of a laser for length measurement by fringe counting,” Metrologia 2, 95–98 (1966).
[CrossRef]

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” Progress in Optics 26, 350–393 (1988).
[CrossRef]

Crisp, D.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Danchi, W. C.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Dandliker, R.

S. Lévéque, Y. Salvadé, R. Dandliker, and O. Scherler,“High-accuracy laser metrology enhances the VLTI,” Laser Focus World 38, 101–104 (2002).

de Bonth, S.

de Groot, P.

de Groot, P. J.

Decker, J. E.

Defrère, D.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Dubovitsky, S.

Eiroa, C.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Eriksson, H.

M. Loefdahl and H. Eriksson, “An algorithm for resolving 2pi ambiguities in interferometric measurements by use of multiple wavelength,” Opt. Eng. 40, 984–990 (2001).
[CrossRef]

Falaggis, K.

K. Falaggis, D. P. Towers, and C. E. Towers, “Theoretical analysis and optimisation of the method of excess fractions for long-range metrology,” Proc. SPIE 7790, 779009 (2010).
[CrossRef]

K. Falaggis, D. P. Towers, and C. E. Towers, “Multiwavelength interferometry: extended range metrology,” Opt. Lett. 34, 950–952 (2009).
[CrossRef] [PubMed]

K. Falaggis, D. P. Towers, and C. E. Towers, “Optimum wavelength selection for the method of excess fractions,” Proc. SPIE 7063, 70630V (2008).
[CrossRef]

Falkowski, P.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Fetterman, H. R.

Gappinger, R. O.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Gillilland, K. E.

K. E. Gillilland, H. D. Cook, K. D. Mielenz, and R. B. Stephens, “Use of a laser for length measurement by fringe counting,” Metrologia 2, 95–98 (1966).
[CrossRef]

Gushov, V. I.

V. I. Gushov and Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
[CrossRef]

Hao, Y.

Hariharan, P.

P. Hariharan, Optical Interferometry, (Academic, 1985).

Henning, T.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Hess, C. F.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324(2002).
[CrossRef]

Hinz, P. M.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Hollis, J. M.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Houairi, K.

Huntley, J. M.

J. M. Huntley and H. O. Saldner, “Error-reduction methods for shape measurement by temporal phase unwrapping,” J. Opt. Soc. Am. A 14, 3188–3196 (1997).
[CrossRef]

J. M. Huntley, “Random phase measurement errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
[CrossRef]

Hunyadi, S. L.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Hurwitz, A.

A. Hurwitz, “Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen,” Acta Math. 12, 367–405(1889).
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Hyland, D. C.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Hyun, S.

S. Hyun, Y.-J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302(2009).
[CrossRef]

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry ,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Jin, J.

S. Hyun, Y.-J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302(2009).
[CrossRef]

J. Jin, Y.-J. Kim, Y. Kim, S.-W. Kim, and C.-S. Kang, “Absolute length calibration of gauge blocks using optical comb of a femtosecond pulse laser,” Opt. Express 14, 5968–5974 (2006).
[CrossRef] [PubMed]

Johnston, K. J.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Jones, J. D. C.

Joo, K.-N.

Kaltenegger, L.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Kang, C.-S.

Kim, S.-W.

Kim, Y.

Kim, Y.-J.

S. Hyun, Y.-J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302(2009).
[CrossRef]

J. Jin, Y.-J. Kim, Y. Kim, S.-W. Kim, and C.-S. Kang, “Absolute length calibration of gauge blocks using optical comb of a femtosecond pulse laser,” Opt. Express 14, 5968–5974 (2006).
[CrossRef] [PubMed]

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry ,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Ksendzov, A.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Labadie, L.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Lane, B. F.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Laughlin, G.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Lawson, P. R.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Lay, O. P.

O. P. Lay, S. Dubovitsky, R. D. Peters, J. P. Burger, S.-W. Ahn, W. H. Steier, H. R. Fetterman, and Y. Chang, “MSTAR: a submicrometer absolute metrology system,” Opt. Lett. 28, 890–892 (2003).
[CrossRef] [PubMed]

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Le Floch, S.

Léger, A.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Lévéque, S.

S. Lévéque, Y. Salvadé, R. Dandliker, and O. Scherler,“High-accuracy laser metrology enhances the VLTI,” Laser Focus World 38, 101–104 (2002).

Lévêque, S.

Li, D.

Li, J.-L.

Liseau, R.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Loefdahl, M.

M. Loefdahl and H. Eriksson, “An algorithm for resolving 2pi ambiguities in interferometric measurements by use of multiple wavelength,” Opt. Eng. 40, 984–990 (2001).
[CrossRef]

Madej, A. A.

Martin, S. R.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Mawet, D.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Mennesson, B.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Michelson, A. A.

A. A. Michelson and J. R. Benoit, “Détermination expérimentale de la valeur du mètre en longueurs d'ondes lumineuses,” Trav. Et Mem. Bur. Int. Poids es Mes. 11, 1–42 (1895).

Mielenz, K. D.

K. E. Gillilland, H. D. Cook, K. D. Mielenz, and R. B. Stephens, “Use of a laser for length measurement by fringe counting,” Metrologia 2, 95–98 (1966).
[CrossRef]

Miles, J. R.

Monnier, J. D.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Nadeborn, W.

W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Osten, W.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324(2002).
[CrossRef]

W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24, 245–260 (1996).
[CrossRef]

Pekelsky, J. R.

Peters, R. D.

O. P. Lay, S. Dubovitsky, R. D. Peters, J. P. Burger, S.-W. Ahn, W. H. Steier, H. R. Fetterman, and Y. Chang, “MSTAR: a submicrometer absolute metrology system,” Opt. Lett. 28, 890–892 (2003).
[CrossRef] [PubMed]

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Pförtner, A.

Quillen, A. C.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Röttgering, H. J. A.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Saldner, H. O.

Salvadé, Y.

Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. 47, 2715–2720 (2008).
[CrossRef] [PubMed]

S. Lévéque, Y. Salvadé, R. Dandliker, and O. Scherler,“High-accuracy laser metrology enhances the VLTI,” Laser Focus World 38, 101–104 (2002).

Scharf, D. P.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Scherler, O.

S. Lévéque, Y. Salvadé, R. Dandliker, and O. Scherler,“High-accuracy laser metrology enhances the VLTI,” Laser Focus World 38, 101–104 (2002).

Schuhler, N.

Schwider, J.

Selsis, F.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Serabyn, E.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Shao, M.

M. Shao, “Astrometry with SIM PlanetQuest,” Proceedings of the International Astronomical Union, Vol.  3 (Cambridge University, 2007), pp. 231–237.
[CrossRef]

Siemsen, K. J.

Siemsen, R. F.

Solodkin, Y. N.

V. I. Gushov and Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
[CrossRef]

Steier, W. H.

Stephens, R. B.

K. E. Gillilland, H. D. Cook, K. D. Mielenz, and R. B. Stephens, “Use of a laser for length measurement by fringe counting,” Metrologia 2, 95–98 (1966).
[CrossRef]

Su, H.-J.

Su, X.-Y.

Surrel, Y.

Y. Surrel, “Two-step temporal phase unwrapping in profilometry” Proc. SPIE 3098, 271–282 (1997).
[CrossRef]

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry ,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Tan, Y.

Temple, S.

Tilford, C. R.

Towers, C. E.

K. Falaggis, D. P. Towers, and C. E. Towers, “Theoretical analysis and optimisation of the method of excess fractions for long-range metrology,” Proc. SPIE 7790, 779009 (2010).
[CrossRef]

K. Falaggis, D. P. Towers, and C. E. Towers, “Multiwavelength interferometry: extended range metrology,” Opt. Lett. 34, 950–952 (2009).
[CrossRef] [PubMed]

K. Falaggis, D. P. Towers, and C. E. Towers, “Optimum wavelength selection for the method of excess fractions,” Proc. SPIE 7063, 70630V (2008).
[CrossRef]

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Generalized multi-frequency selection for full-field interferometric shape measurement,” Proc. SPIE 5502, 406–409 (2004).
[CrossRef]

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Optimum frequency selection in multifrequency interferometry,” Opt. Lett. 28, 887–889 (2003).
[CrossRef] [PubMed]

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Time efficient Chinese remainder theorem algorithm for full-field fringe phase analysis in multi-wavelength interferometry,” Opt. Express 12, 1136–1143.
[CrossRef] [PubMed]

Towers, D. P.

K. Falaggis, D. P. Towers, and C. E. Towers, “Theoretical analysis and optimisation of the method of excess fractions for long-range metrology,” Proc. SPIE 7790, 779009 (2010).
[CrossRef]

K. Falaggis, D. P. Towers, and C. E. Towers, “Multiwavelength interferometry: extended range metrology,” Opt. Lett. 34, 950–952 (2009).
[CrossRef] [PubMed]

K. Falaggis, D. P. Towers, and C. E. Towers, “Optimum wavelength selection for the method of excess fractions,” Proc. SPIE 7063, 70630V (2008).
[CrossRef]

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Generalized multi-frequency selection for full-field interferometric shape measurement,” Proc. SPIE 5502, 406–409 (2004).
[CrossRef]

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Optimum frequency selection in multifrequency interferometry,” Opt. Lett. 28, 887–889 (2003).
[CrossRef] [PubMed]

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Time efficient Chinese remainder theorem algorithm for full-field fringe phase analysis in multi-wavelength interferometry,” Opt. Express 12, 1136–1143.
[CrossRef] [PubMed]

Traub, W. A.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Unwin, S. C.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Wilner, D. J.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Cambridge University, 2006), Chap. 7.6.

Woolf, N. J.

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

Wyant, J. C.

Ye, J.

Zhang, Y.

Zhao, H.

Zhao, Y.

Zhong, J.

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A. Hurwitz, “Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen,” Acta Math. 12, 367–405(1889).
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J.-L. Li, H.-J. Su, and X.-Y. Su, “Two-frequency grating used in phase-measuring profilometry,” Appl. Opt. 36, 277–280(1997).
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Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. 47, 2715–2720 (2008).
[CrossRef] [PubMed]

J. E. Decker, J. R. Miles, A. A. Madej, R. F. Siemsen, K. J. Siemsen, S. de Bonth, K. Bustraan, S. Temple, and J. R. Pekelsky, “Increasing the range of unambiguity in step-height measurement with multiple-wavelength interferometry-application to absolute long gauge block measurement,” Appl. Opt. 42, 5670–5678 (2003).
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[CrossRef] [PubMed]

H. Zhao, W. Chen, and Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33, 4497–4500 (1994).
[CrossRef] [PubMed]

P. de Groot, “Design of error-compensating algorithms for sinusoidal phase shifting interferometry,” Appl. Opt. 48, 6788–6796 (2009).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Y. Hao, Y. Zhao, and D. Li, “Multifrequency grating projection profilometry based on the nonlinear excess fraction method,” Appl. Opt. 38, 4106–4110 (1999).
[CrossRef]

A. Pförtner and J. Schwider, “Red-green-blue interferometer for the metrology of discontinuous structures,” Appl. Opt. 42, 667–673 (2003).
[CrossRef] [PubMed]

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry ,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

J. Opt. Soc. Am. A

Laser Focus World

S. Lévéque, Y. Salvadé, R. Dandliker, and O. Scherler,“High-accuracy laser metrology enhances the VLTI,” Laser Focus World 38, 101–104 (2002).

Meas. Sci. Technol.

S. Hyun, Y.-J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20, 095302(2009).
[CrossRef]

Metrologia

K. E. Gillilland, H. D. Cook, K. D. Mielenz, and R. B. Stephens, “Use of a laser for length measurement by fringe counting,” Metrologia 2, 95–98 (1966).
[CrossRef]

Opt. Eng.

M. Loefdahl and H. Eriksson, “An algorithm for resolving 2pi ambiguities in interferometric measurements by use of multiple wavelength,” Opt. Eng. 40, 984–990 (2001).
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V. I. Gushov and Y. N. Solodkin, “Automatic processing of fringe patterns in integer interferometers,” Opt. Lasers Eng. 14, 311–324 (1991).
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C. E. Towers, D. P. Towers, and J. D. C. Jones, “Generalized multi-frequency selection for full-field interferometric shape measurement,” Proc. SPIE 5502, 406–409 (2004).
[CrossRef]

K. Falaggis, D. P. Towers, and C. E. Towers, “Optimum wavelength selection for the method of excess fractions,” Proc. SPIE 7063, 70630V (2008).
[CrossRef]

K. Falaggis, D. P. Towers, and C. E. Towers, “Theoretical analysis and optimisation of the method of excess fractions for long-range metrology,” Proc. SPIE 7790, 779009 (2010).
[CrossRef]

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[CrossRef]

P. R. Lawson, O. P. Lay, K. J. Johnston, C. A. Beichman, S. C. Unwin, S. R. Martin, S. L. Hunyadi, O. Absil, R. L. Akeson, J. Bally, P. Bordé, D. Crisp, W. C. Danchi, D. Defrère, C. Eiroa, P. Falkowski, T. Henning, P. M. Hinz, J. M. Hollis, D. C. Hyland, L. Kaltenegger, L. Labadie, B. F. Lane, G. Laughlin, A. Léger, R. Liseau, D. Mawet, B. Mennesson, J. D. Monnier, A. C. Quillen, H. J. A. Röttgering, F. Selsis, E. Serabyn, D. J. Wilner, N. J. Woolf, W. A. Traub, R. O. Gappinger, A. Ksendzov, R. D. Peters, and D. P. Scharf, “Terrestrial Planet Finder Interferometer (TPF-I) Whitepaper for the AAAC Exoplanet Task Force,” http://planetquest1.jpl.nasa.gov/TPF-I/TPFIwhitepaper.pdf.

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Figures (4)

Fig. 1
Fig. 1

Absolute residual error pattern for a two-wavelength system with a scaling factor of s f = 7.88 .

Fig. 2
Fig. 2

Absolute residual error and the estimated lower boundary for a two-wavelength system with a scaling factor of s f = 7.88 . The minimum value of the residual error is highlighted for the exemplary cases with k = 0 , k = 1 , and k = 2 .

Fig. 3
Fig. 3

Extended UMR (solid curve) of a two-wavelength system with λ 0 = 405 nm and a phase noise of 1 / 1000 th of a fringe with (a)  k max = 2 , and (b) with an unbounded value of k max . The corresponding noise limited value is indicated by the dotted line.

Fig. 4
Fig. 4

Values of the residual error for various values of m 0 for s f = S s f s f = 25.4 and α = 7 / 13 with ( Q α W α ) = 13 . The residual error shown corresponds to the fractional fringe orders ε 0 = 0.3706 , ε 1 = 0.4199 , and ε 2 = 0.3245 .

Equations (100)

Equations on this page are rendered with MathJax. Learn more.

OPD = u 0 λ 0 , OPD = u 1 λ 1 , , OPD = u N 1 λ N 1 ,
OPD = ( M 0 i + E 0 i ) Λ 0 i ,
Λ 0 i = λ i λ 0 λ i λ 0 .
m i = ( m 0 + ε 0 ) λ 0 λ i ε i .
r i ( m 0 ) = fract ( ( m 0 + ε 0 ) λ 0 λ i ε i ) ,
R ( m 0 ) = i = 1 N 1 | r i ( m 0 ) | 2 .
r i ( m 0 ) = fract [ ( m 0 + ε 0 ) λ 0 Λ 0 i + E 0 i ] .
r i ( m 0 ) = fract [ ( m 0 + ε 0 ) s f i + E 0 i ] .
σ R ( N 1 ) ( σ ε 0 ) 2 + 1 R 2 i = 1 N 1 ( r i ( ε 0 , ε i ) ) 2 ( σ ε i ) 2 .
r 1 ( m 0 = + 1 ) = fract [ 1 s f ] .
σ R ( σ ε 0 ) 2 + ( σ ε 1 ) 2 ,
| fract [ 1 s f ] | 6 2 σ ε ,
s f 1 / ( 6 2 σ ε ) .
m 0 = [ 0 , floor ( s f ) 1 ] ,
m 0 = NINT ( s f ) ,
r 1 ( m 0 = NINT ( s f ) ) = fract [ ( NINT ( s f ) + ε 0 ) 1 s f + E 01 ] .
| r 1 ( m 0 = NINT ( s f ) ) | = | fract [ ( NINT ( s f ) ) 1 s f ] | | r 1 ( m 0 = NINT ( s f ) ) | = | fract [ ( s f fract ( s f ) s f ] | = | fract [ 1 s f fract ( s f ) ] | ,
| r 1 ( m 0 = NINT ( s f ) ) | = | 1 s f fract ( s f ) | .
| 1 s f fract ( s f ) | 6 2 σ ε ,
s f 1 6 2 σ ε | fract ( s f ) | .
R ( m 0 ) = | fract [ m 0 s f ( ε 0 s f ( ε 0 ε 1 ) ) ] | = | fract [ m 0 s f + f ( ε 0 , ε 1 ) ] | ,
f ( ε 0 , ε 1 ) = ( ε 0 s f ( ε 0 ε 1 ) ) .
R ( m 0 ) = | fract [ m 0 s f ] | .
x 0 = 1 s f s k q k < 1 ,
R ( m 0 ) | x 0 p k + 1 | ,
S = | s k + 1 | Q = | q k + 1 | if     { ! k N | | 1 / V | < 6 σ R | 1 / T | } ,
T = p k + 1 / x 0 , V = p k + 2 / x 0 , W = | q k | for     fract ( x k + 1 ) 0 T = p k + 1 / x 0 , V = + , W = 0 for     fract ( x k + 1 ) = 0 , ,
| R ( m 0 ) | { | 1 T | for     m 0 = [ | q k | , ( | q k + 1 | | q k | ) 1 ] | 1 T | | 1 V | for     m 0 = [ ( | q k + 1 | | q k | ) , ( | q k + 1 | 1 ) ] | 1 V | for     m 0 = | q k + 1 | ,
UMR = ( Q W ) λ 0 ,
m 0 = [ 0 , ( Q W 1 ) ] .
ψ s f = | 1 T | .
R ( m 0 ) = | fract ( L ( m 0 ) Λ 02 + E 02 ) | 2 + | fract ( L ( m 0 ) Λ 01 + E 01 ) | 2 = | fract ( m 0 + ε 0 s f + ( ε 0 ε 2 ) ) | 2 + | fract ( ( 1 α ) m 0 + ε 0 s f + ( ε 0 ε 1 ) ) | 2 ,
σ R ( σ ε 0 ) 2 + 1 R 2 i = 1 2 ( r i ( ε 0 , ε i ) ) 2 ( σ ε i ) 2 ,
R ( m 0 = + 1 ) = 1 s f 1 + ( 1 α ) 2 .
δ = 1 s f 1 + ( 1 α ) 2 .
δ 6 σ R ,
m 0 = NINT ( n S s f s f ) = n S s f s f fract ( n S s f s f ) ,
R ( m 0 = NINT ( n S s f s f ) = ρ [ n S s f , s f ] 2 + | fract [ ( 1 α ) n S s f ( 1 α ) ρ [ n S s f , s f ] ] | 2 ,
| fract [ ( 1 α ) n S s f ] ( 1 α ) ρ [ n S s f , s f ] | < 0.5.
R ( m 0 = NINT ( n S s f s f ) = ρ [ n S s f , s f ] 2 + ( fract [ n ( 1 α ) S s f ] ( 1 α ) ρ [ n S s f , s f ] ) 2 .
R ( m 0 = NINT ( n S s f s f ) = | η [ n , ( 1 α ) S s f ] | ,
η [ n , ( 1 α ) S s f ] = fract [ n ( 1 α ) S s f ] = fract [ n fract [ ( 1 α ) S s f ] ] = fract [ n / s f α ] ,
( s f ) α = 1 / fract [ ( 1 α ) S s f ] .
x 0 = 1 s f α s k q k < 1 .
UMR = floor [ ( Q α W α ) S s f s f W s f ] λ 0 ,
m 0 = [ 0 , floor [ ( Q α W α ) S s f s f W s f ] 1 ] .
UMR = floor [ ( 13 0 ) × 1 × 25.4 1 ] λ 0 = 329 λ 0 .
| R ( b , t ) | = { ( t δ ) 2 for     b = 0 ( b ψ 0 ) 2 + ( γ ( b ) t δ ) 2 for     b 0 ,
ψ ( b ) = η [ n , 1 / s f a ] 1 + 1 / tan 2 θ ,
ψ 0 = ± 1 T α 1 1 + ( 1 α ) 2 ,
| R | = { ψ 0 2 + γ 2 for     γ δ / 2 ψ 0 2 + ( δ γ ) 2 for     γ > δ / 2 ,
| ψ 0 | 6 σ R ,
S α = | s k + 1 | Q α = | q k + 1 | if     { ! k N | | 1 / V α | < 6 σ R 1 + ( 1 α ) 2 | 1 / T α | } ,
T α = p k + 1 / x 0 , V α = p k + 2 / x 0 , W α = | q k | for     fract ( x ( k + 1 ) ) 0 T α = p k + 1 / x 0 , V α = + , W α = 0 for     fract ( x ( k + 1 ) ) = 0 . ,
R ( m 0 ) = + | fract ( ( 1 α 1 ) m 0 s f ) | 2 | fract ( m 0 s f ) | 2 + | fract ( ( 1 α 1 ) m 0 s f ) | 2 .
δ = 1 s f 1 + ( 1 α 1 ) 2 + ( 1 α 2 ) 2 ,
δ 6 σ R .
σ R 3 ( σ ε 0 ) 2 + 1 R 2 i = 1 3 ( r i ( ε 0 , ε i ) ) 2 ( σ ε i ) 2 ,
R ( m 0 = NINT ( n S s f s f ) ) = + ( η [ n , 1 / s f a 2 ] ( 1 α 2 ) ρ [ n S s f , s f ] ) 2 + ( η [ n , 1 / s f a 1 ] ( 1 α 1 ) ρ [ n S s f , s f ] ) 2 ρ [ n S s f , s f ] 2 ,
1 / s f α 2 = fract [ ( 1 α 2 ) S s f ] .
R ( m 0 = NINT ( n S s f s f ) = ( η [ n , 1 / s f a 2 ] ) 2 + ( η [ n , β 1 / s f a 2 ] ) 2 = ( fract [ n / s f a 2 ] ) 2 + ( fract [ β 1 n / s f a 2 ] ) 2 ,
ψ α 2 = η [ n 0 , 1 / s f a 2 ] 1 + ( 1 α 2 ) 2 ,
| ψ α 2 | 6 σ R .
n = NINT ( c 1 S α 2 s f α 2 ) ,
R ( n = NINT ( c 1 S α 2 s f α 2 ) ) = ( fract [ n / s f a 2 ] ) 2 + ( fract [ β 1 n / s f a 2 ] ) 2 = ( fract [ c 1 S α 2 ρ [ c 1 S α 2 , s f α 2 ] ] ) 2 + ( fract [ β 1 c 1 S α 2 β 1 ρ [ c 1 S α 2 , s f α 2 ] ] ) 2 ,
R ( n = NINT ( c 1 S α 2 s f α 2 ) ) ( fract [ c 1 S α 2 ] ) 2 + ( fract [ β 1 c 1 S α 2 ] ) 2 | fract [ β 1 c 1 S α 2 ] | ,
s f β 1 = 1 / fract [ β 1 S α 2 ] .
ψ β 1 = η [ c 1 , 1 / s f β 1 ] 1 + β 1 2 ,
| ψ β 1 | 6 σ R .
S β = | s k + 1 | Q β = | q k + 1 | if     { ! k N | | 1 / V β | < 6 σ R 1 + β 1 2 | 1 / T β | } ,
T β = p k + 1 / x 0 , V β = p k + 2 / x 0 , W β = | q k | for     fract ( x ( k + 1 ) ) 0 T β = p k + 1 / x 0 , V β = + , W β = 0 for     fract ( x ( k + 1 ) ) = 0 ,
δ = 1 s f 1 + i = 1 N 2 ( 1 α i ) 2 ,
ψ s f = | 1 T s f | ,
s f α = fract [ ( 1 α N 2 ) S s f ] , s f β ( N 3 ) = fract [ β ( N 3 ) S α ] , s f β ( N 2 ) = fract [ β ( N 2 ) S β ( N 3 ) ] , s f β 1 = fract [ β 1 S β 2 ] ,
ψ α = 1 T α 1 1 + ( 1 α N 2 ) 2 ,
ψ β n = 1 T β n 1 1 + ( β n ) 2 ,
β n = 1 α n 1 α n + 1 ,
min { δ , ψ s f , ψ α ( N 2 ) , ψ β ( N 3 ) , ... , ψ β 1 } 6 σ R .
δ 6 σ R ,
{ 1531 , 1543 , 1549 , 1553 , 1559 , 1567 , 1571 , 1579 } .
x 0 = a 0 1 x 1 , x 1 = a 1 1 x 2 , , x k = a k 1 x k + 1 , ,
x 0 = a 0 + 1 x 1 , x 1 = a 1 + 1 x 2 , , x k = a k + 1 x k + 1 , ,
x 0 = a 0 + 1 x 1 = a 0 + 1 a 1 + 1 x 2 = a 0 + 1 a 1 + 1 a 2 + 1 x 3 , etc .
x 0 = ( a 0 , a 1 , a 2 , ... , a k , x k + 1 ) ,
x 0 s k q k = ( a 0 , a 1 , a 2 , ... , a k ) ,
s k = a k s k 1 + s k 2 , and q k = a k q k 1 + q k 2 ,
s k 1 q k q k 1 s k = ( 1 ) k ,
x 0 = s k x k + 1 + s k 1 q k x k + 1 + q k 1 .
x 0 s k q k = ( 1 q k ) 2 ( 1 ) k ( x k + 1 + q k 1 q k ) .
fract [ ( x 0 s k q k ) q k ] = x 0 p k + 1 ( 1 ) k + 1 ,
fract [ ( 1 x 0 q k s k ) s k ] = ( 1 ) k p k + 1 ,
p k = j = 0 j = k x j .
R ( m 0 ) = | fract ( m 0 / s f ) | ,
x 0 = 1 s f ,
m 0 = c 1 | q k | + c 2 | q k + 1 | ,
R ( m 0 = c 1 | q k | + c 2 | q k + 1 | ) = | fract ( c 1 | q k | x 0 + c 2 | q k + 1 | x 0 ) | ,
R ( m 0 = c 1 | q k | + c 2 | q k + 1 | ) = | fract ( c 1 sign [ q k ] x 0 p k + 1 ( 1 ) k + 1 + c 2 sign [ q k + 1 ] x 0 p k + 2 ( 1 ) k ) | ,
sign ( x ) = { + 1 for     x > 0 0 for     x = 0 1 for     x < 0 .
R ( m 0 = c 1 | q k | + c 2 | q k + 1 | ) = x 0 / p k + 1 ( for     m 0 = | q k | ) , R ( m 0 = c 1 | q k | + c 2 | q k + 1 | ) = x 0 | s i g n ( q k + 1 ) p k + 2 s i g n ( q k ) p k + 1 | ( for     m 0 = | q k + 1 | | q k | ) , R ( m 0 = c 1 | q k | + c 2 | q k + 1 | ) = x 0 / p k + 2 ( for     m 0 = | q k | ) ,
R ( m 0 = c 1 | q k | + c 2 | q k + 1 | ) x 0 / p k + 1 ( for     m 0 = | q k | ) , R ( m 0 = c 1 | q k | + c 2 | q k + 1 | ) | x 0 p k + 2 x 0 p k + 1 | ( for     m 0 = | q k + 1 | | q k | ) , R ( m 0 = c 1 | q k | + c 2 | q k + 1 | ) x 0 / p k + 2 ( for     m 0 = | q k + 1 | ) .

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